The fate of an immigration overhaul in the House has become a guessing game, but in two separate statements this week, one Republican lawmaker with significant influence over how the process will unfold hinted at what could — or won't — come to pass.

First, in an interview on Monday with conservative radio host Hugh Hewitt, Judiciary Chairman Robert W. Goodlatte, R-Va., said he supported providing legal status for "Dreamers," the children of undocumented immigrants who were brought to the country illegally by their parents. He wouldn't, however, support a pathway to citizenship, which Democrats say is a sticking point for them in any comprehensive overhaul of the nation's immigration system. Goodlatte said:

"If you were to do something, I would start first of all with children who were brought here illegally by their parents. They’ve grown up here. They’ve been educated here. They are ready to face the world and they have no documents. I think there’s a more compelling argument to be made for them. But, even for them, I would say that they get a legal status in the United States and not a pathway to citizenship that is created especially for them. In other words, they get that legal status if they have an employer who says I’ve got a job which I can’t find a U.S. citizen and I want to petition for them, ah, they can do that, but I wouldn’t give them the pathway to a Green Card and ultimately citizenship based simply on their entering the country illegally."

On Wednesday afternoon, Goodlatte sent a letter to Secretary of Homeland Security Janet Napolitano weighing in on another aspect of the immigration debate: reports of a surge of undocumented immigrants coming into the country claiming asylum under false pretenses "to game the system by getting a free pass into the U.S. and a court date that they do not plan to show up for."

Goodlatte and other Republicans have said that any immigration overhaul effort must not be tantamount to an "amnesty" bill for the 11 million undocumented citizens living in the United States. Underscoring this, some Republicans have pledged not to support any bill that provides a pathway to citizenship or legal status until legislation is first passed to secure the border.

Some Republicans say that now is the not the time to pass an immigration bill because President Barack Obama has indicated his unwillingness to uphold the law and respect the checks and balances of Congress.

Goodlatte signaled in his letter to Napolitano that he shared some of these concerns about the White House's trustworthiness, noting administration inconsistencies with how the law is followed and a pattern of using executive actions to ignore the law.

"Once again the Administration has chosen to turn the immigration enforcement switch off in a manner contrary to the intent of Congress, by simply enforcing the immigration laws when, where, and as it is deemed fit," Goodlatte wrote. "Such actions are the primary reason why our immigration system is broken today. We plan to conduct oversight of this issue and address concerns via the House’s step-by-step approach to reforming our immigration system."